Hagfors' law revisited

Authors


Abstract

[1] Hagfors' scattering law, σ°H(θ), is in wide use in connection with the study of backscatter data from planetary surfaces because it provides good agreement with a variety of observations. The surface root-mean-square slope inferred on the basis of σ°H(θ) is customarily taken as C−1/2, where C is the shape parameter in σ°H(θ). The relationship between the surface slope and C is indefinite, however, because of the indeterminateness of the surface scales contributing to the scattering process. Moreover, the horizontal scale of the inferred slope obtained is not specified. As a consequence of limitations in the Kirchhoff approximation on which it is predicated, σ°H(θ) does not conserve energy. The use of a fractional Brownian fractal surface model leads to a scattering law with the same functional form as σ°H(θ) when the Hurst exponent characterizing the fractal model is 1/2. Fractal-based scattering laws, derived by applying the Kirchhoff approximation, suffer the same deficiency with regard to conservation of energy. In contrast to σ°H(θ), slope information for fractal-based laws is explicit with respect to horizontal scale. Both σ°H(θ) and fractal-based laws require that the illuminated surface area exceeds a certain value, which is a function of the electromagnetic wavelength and surface parameters, in order to reduce the surface radar cross section overestimation error, introduced by a mathematical approximation, below some specified value. This requirement may be necessary to take into account in experiments where the radar resolution cells are comparable in size to the wavelength, such as in Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS).

1. Introduction

[2] Analysis of radio wave scattering from planetary surfaces allows the inference of statistical surface properties on a range of relevant structural scales that are conditioned by the wavelength of observations. Scattering laws describe the variation of the radar cross section (RCS) with the angles of incidence and scattering in the bistatic case, and with the angle of incidence in the monostatic, or backscatter, case. Since scattering laws are typically parameterized by physical surface properties, such as the Fresnel reflection coefficient, the surface root-mean-square (RMS) height, and the surface RMS slope, they provide a means for estimating such surface parameters by comparison of theory with observations. Knowledge of these parameters can be used to constrain the composition, structure, and, in some instances, the natural history of the surface.

[3] The eponymic backscattering law developed by Hagfors [1964], which we denote σ°H(θ), has become widely used in planetary radar studies because it is, in many instances, in close agreement with data [e.g., Simpson and Tyler, 1982]. The width parameter “C” of σ°H(θ) provides a measure of the angular extent of the backscattering lobe and hence the roughness of the surface. In addition, the backscatter strength at normal incidence to the mean surface is given by ρC/2, where ρ is the power reflection coefficient. In this work we discuss some theoretical aspects of σ°H(θ) pertinent to the analysis of remote sensing data.

[4] Notwithstanding its success in matching observational data, the use of σ°H(θ) has been subject to several objections. The most notable of these, that use of the exponential correlation function is entirely inappropriate on both mathematical and physical grounds, was raised by Barrick [1970] whose critique is addressed in several places in this paper. Although the central issues have remained unresolved in the literature, the controversy over the adequacy of σ°H(θ) has subsided. It can be said that σ°H(θ) is now the de facto scattering law employed in most surface studies of the terrestrial planets, and that Hagfors' parameter C is the standard measure of surface roughness applied to the terrestrial planets and other similarly scattering bodies.

[5] The derivation of σ°H(θ) assumes a classically modeled surface of homogeneous material with Gaussian height distribution and an exponential correlation function. The Kirchhoff approximation (KA) is used to estimate the fields induced on the surface by the incident electromagnetic radiation. Hence Hagfors' result is restricted by the limitations of KA, among other things. This is manifest, for example, in the failure of σ°H(θ) to conserve energy because KA cannot capture the full range of surface scattering behaviors, as discussed below.

[6] In analyses involving the use of σ°H(θ) to represent the variation of backscatter RCS with angle, the inverse of the square root of the width parameter C of σ°H(θ) is usually taken as the RMS slope of the surface, following a derivation by Hagfors [1970]. We discuss the basis for this result by investigating the scattering integral leading to σ°H(θ). We show how changing the underlying assumption in a reasonable way yields different relations between the width parameter C and the surface RMS slope, thereby illustrating an indefiniteness inherent in the value of the surface RMS slope inferred on the basis of σ°H(θ).

[7] Fractal geometry has been shown to provide more accurate representations of natural surfaces than do standard stochastic models [e.g., Mandelbrot, 1982; Voss, 1985; Falconer, 1990; Barnsley, 2000]. Consequently, fractal geometry has become increasingly attractive both as a model for physical surfaces and as the basis for a new generation of radio wave scattering models. In particular, the use of the fractional Brownian (FB) fractal surface model and the Kirchhoff approximation gives a scattering law with the same functional form as σ°H(θ) when the Hurst exponent characterizing the model equals 1/2. Fractal-based scattering laws parameterized by the Hurst exponent yield a different result for the surface RMS slope which appears more physically sound than does the C−1/2 result now in common use. In contrast with the classical result, the fractal-based model leads to an RMS slope estimate explicitly linked to horizontal surface scales. This latter result is useful because the explicit variation of surface parameters with the scale at which they are measured or estimated is a fundamental characteristic of natural surfaces. Furthermore, it leads to scale-dependent scattering properties that correspond to certain aspects of planetary radar observations. Fractal-based laws, however, also fail to conserve energy as a result of employing KA to represent the boundary conditions.

[8] The remainder of this paper is organized as follows. In section 2, we provide a general discussion of scattering laws and their derivation; the Kirchhoff approximation and its reduction to σ°H(θ) are summarized. Section 3 is devoted to an analysis of the integral leading to σ°H(θ). We emphasize the surface scales that contribute most strongly to the value of the scattering integral and hence comprise the physical range of surface scales that dominate the scattering process, granted the validity of the several assumptions involved in the derivation of σ°H(θ). In section 4, the conservation of energy aspects of σ°H(θ) are studied by recourse to the bistatic version. We show that the law does not conserve energy following an analysis provided by Barrick [1970]. In section 5, we compare σ°H(θ) with scattering laws derived on the basis of the FB surface models emphasizing the difference in the treatment of surface scales in the two approaches. As with σ°H(θ), the fractal-based laws require a minimum illuminated surface size in order that the estimation error in the value of the surface RCS, resulting from a mathematical approximation, not exceed a specified value. A numerical example pertinent to the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) illustrates this requirement. Section 6 contains our conclusions.

2. Kirchhoff Approximation and Hagfors' Law

[9] Two distinct components of analysis must be combined in developing a scattering law: (1) a surface model and (2) a technique for solution of the electromagnetic (EM) wave-surface interaction. Widely used conventional surface models include statistically stationary surfaces with Gaussian height distribution and a specified correlation function which can have, for instance, either a Gaussian or other form. The EM technique describes how the incident wave interacts with the surface, and how fields from induced currents on or within the surface are reradiated, thereby giving rise to the scattered signal.

[10] The Kirchhoff approximation (KA) is a widely used technique for solving surface scattering problems due to its relative simplicity and practicality, and is demonstrably sound when applied appropriately. Application of KA “solves” the EM boundary condition problem by employing a physical optics approach in which the surface boundary fields at each point are obtained by replacing the local surface with an infinite tangent plane [e.g., Beckmann and Spizzichino, 1963; Ogilvy, 1991; Tsang et al., 2000].

[11] Despite its widespread use, KA is valid, i.e., KA produces accurate results, only for a certain range of surface and EM parameters. When applied to estimation of surface fields on rough surfaces, a consequence of the tangent plane approximation is that the surface radius of curvature is a critical parameter controlling the validity of KA. Since the surface is treated locally as an infinite plane in order to obtain the surface boundary conditions, the local radius of curvature should intuitively be no less than approximately the EM wavelength, λ, lest the actual surface deviate grossly from planar rendering the physical optics approximation inapplicable due to a combination of phase and amplitude errors. Rather than using the radius of curvature of the mathematical surface, Tyler [1976] (for classical surfaces) and Sultan-Salem and Tyler [2004] (for fractal surfaces) propose the use of an “effective” wavelength-dependent radius of curvature 〈rc(λ)〉 that takes into account the surface-wave interaction. Solving the inequality 〈rc(λ)〉 > λ yields an estimate of the region in the surface-EM space where the application of KA is valid. Of importance to the main topic of this paper is the point that KA, on which σ°H(θ) is based, is of limited applicability.

[12] Analytically, when an expression for the scattered field Es is obtained in terms of the scattering surface, one can proceed in two ways. By computing 〈Es〉, i.e., the statistical average of the scattered field over surface realizations, one can then square the result to calculate the coherent RCS,

equation image

where the subscript “sm” stands for “square mean,” Ei is the complex amplitude of the incident field, and r is the distance from the surface to the point of observation. Alternatively, one can compute the total cross section by squaring Es and then averaging over surface realizations, that is,

equation image

where the subscript “ms” stands for “mean square.” Observationally, σsm and σms can be distinguished on the basis of signal processing in the radio wave receiver. Most radars and similar systems sense σms, or sometimes a combination of σsm and σms. In the analysis below the specific bistatic RCS, which is the bistatic σms normalized by the geometric scattering area, is denoted by σ°bi. The corresponding specific monostatic or backscatter RCS is denoted by σ°.

[13] The well-known Hagfors' [1964] scattering law follows from the application of KA to a conventional stochastic Gaussian surface with exponential correlation function. The exponential correlation function has the form: exp(−R), where λ is the correlation length and the scale R is the horizontal distance between two surface points.

[14] In order to understand how the expression of σ°H(θ) is obtained, assume the scattering geometry in Figure 1 with a plane wave incident, and that A = sin(θi) − sin(θs)cos(ϕs), B = −sin(θs)sin(ϕs), and D = −[cos(θi) + cos(θs)]. Following Ogilvy [1991], the resulting integral for the specific bistatic RCS is

equation image

where the variable of integration R, as before, is the horizontal separation between two points on the surface, λ is the EM wavelength, ρ is the Fresnel power reflection coefficient, F is a function of θi, θs, ϕs and the polarization of the transmitting and receiving antennas [see Ogilvy, 1991, pp. 128–129], J is the zeroth-order Bessel function, and hrms is the RMS surface height. In this formulation, the dielectric nature of the surface is incorporated in the multiplicative factor ρ, and the effects of shadowing and multiple scattering are neglected.

Figure 1.

Scattering coordinate system. The incident wave unit vector equation imagei = sin(θi)equation image − cos(θi)equation image; that is, the incident wave is in the x-z plane with an angle θi with the z axis. The scattered wave unit vector equation images = sin(θs)cos(ϕs)equation image + sin(θs)sin(ϕs)equation image + cos(θs)equation image, where θs is the polar angle measured from the z axis and ϕs is the azimuthal angle about the z axis from the x-z plane.

[15] In the backscattering case θs = θi = θ and ϕs = π, yielding

equation image

where F for backscatter is equal to 2/cos(θ) for the expected sense of polarization, i.e., the same component for linear polarization and the opposite rotational sense component for circular polarization. For the orthogonally polarized component, i.e., the perpendicular component for linear polarization and the same rotational sense component for circular polarization, F is equal to zero indicating that in this mathematical formulation KA predicts no depolarization in the backscatter direction. Evaluating integral (4) yields Hagfors' backscatter law,

equation image

where the width parameter C is given by

equation image

The effect of setting the upper limit of the integrals (3) and (4) to ∞, as though the illuminated surface were infinite in extent, is discussed in the next section.

[16] Figure 2 shows the backscatter RCS given by σ°H(θ) as a function of the angle of incidence. Parameter C is related to the roughness of the surface relative to the EM wavelength used to probe the surface. As is clear from (6), C decreases as hrms/λ increases, i.e., as the surface roughens with respect to λ. Figure 2 illustrates the variation of σ°H(θ) for C = 50, 250, and 1250 with ρ = 1.

Figure 2.

Hagfors' scattering law, σH°(θ), which gives the specific backscatter RCS as a function of the angle of incidence. Three plots here correspond to C = 50, 250, and 1250, with ρ = 1. The larger the value of C, the smoother the surface with respect to the EM wavelength.

3. Horizontal Scales and Surface RMS Slope

[17] We can obtain some insight into the behavior of σ°H(θ) and its relationship to the underlying surface scales that contribute most strongly to scattering by examining the behavior of the integrand in (4). The role of scale enters expression (4) through the variable of integration R and the limits placed on the integral. Since R is the horizontal separation between two points on the surface, every interval [R, R + dR] represents the contribution from a distinct surface scale. Near normal incidence, cos(θ) ≈ 1 and the Bessel function J(.) ≈ 1, with the result that the integrand is proportional to R exp(−aR) where a = (16π2hrms2)/(λλ2) = (4π)/(λequation image), as is evident from (4) and (6). The shape of the integrand R exp(−aR) is shown in Figure 3; the maximum of the curve occurs at R = 1/a.

Figure 3.

Integrand of (4) for θ ≈ 0 scaled by parameter a = (16π2hrms2)/(λλ2). Value of Rmin sets the minimum scale significantly contributing to the scattering process. Rmin is chosen such that the area under curve within the interval [0, Rmin] is small compared to the total area. Value of Rmax corresponds to the radial extent of the illuminated surface.

[18] Let Rmax be the extent of the illuminated surface in some specific observation, and let the ratio of the area under R exp(−aR) within the interval [Rmax, ∞] to the total area under the curve be y, which can easily be shown to equal (1 + aRmax) exp (−aRmax). The use of σ°H(0) overestimates the specific backscatter RCS by the fraction y/(1 − y) because the upper integration limit in (4) is set to ∞ while the illuminated surface is limited to Rmax. Restricting this overestimation error at normal incidence to be less than 1/10, for example, requires that y < 1/11. Numerically solving the inequality (1 + aRmax) exp (−aRmax) = y < 1/11 for Rmax, we find,

equation image

[19] Given a surface with fixed dimension Rmax initially at the lower limit of inequality (7), it follows that the inequality is increasingly violated as, say, the surface is smoothed by decreasing hrms/λ. As a consequence, use of σ°H(0) increasingly overestimates the surface specific RCS. Fulfilling the condition given by (7), which links the surface dimension, correlation length, RMS height, and the EM wavelength, justifies the infinite bound for the upper limit of integral (4) if the overestimation error is required to be less than some specified value, here 1/10 for illustration. It is important to note that because of energy conservation issues that are discussed in the next section, the overestimation error mentioned here does not mean necessarily that use of σ°H(θ) actually overestimates the physical RCS, since the two effects tend to offset one another. Also, the issue of Rmax is particularly relevant in cases where the radar resolution cells have a size Rmax that is comparable to the EM wavelength probing the surface. This is evident from (7) where Rmax/λ is required to exceed equation image/π to minimize the overestimation error. If λ is much smaller than Rmax, then inequality (7) is readily satisfied over the practical range of values of parameter C.

[20] We now address the following problem posed by Barrick [1970]. It is clear from (5) and (6) that decreasing λ decreases both C and σ°H(0), the latter of which is equal to ρC/2. Contrast this with the well-known result from the theory of scattering from perfectly conducting smooth surfaces, i.e., ρ = 1 and hrms = 0, where σ° = 4πA2 in the case of normal incidence. According to this formula, as λ decreases, σ°(0) increases contrary to the behavior exhibited by σ°H(0). Barrick (p. 648) concluded that “the result [of using an exponential correlation function] exhibits the wrong wavelength behavior in the high frequency limit.”

[21] This problem does not emerge because of the use of the exponential correlation function per se, however. Rather, it stems from letting the upper limit of integration in (4) increase without bound while the probed surface is finite in extent. This last point is illustrated by assuming a smooth perfectly conducting circular surface with radius (A/π)1/2, and by setting the upper limit of integration in expression (4) to this radius. The result is

equation image

which is the expected value. For a smooth surface, decreasing λ means an increase in the surface area relative to λ and thus an increase in σ°(0). On the other hand, for rough surfaces to which σ°H(θ) is applicable, σ°H(0)(=ρC/2 ∝ λ2) decreases with λ, in agreement with observations. The reason is that for natural rough surfaces and typical probing frequencies, surface roughness on decreasingly small scales plays a larger contributory role in scattering as the EM wavelength decreases with the result that scattered power is increasingly removed from the quasi-specular direction.

[22] Consider now the effect of roughness corresponding to small values of R. Choosing Rmin ∈ [0, 1/a], it is clear from Figure 3 that the area under the curve within the interval [0, Rmin] is small when compared with the total area. Given this, it can be hypothesized that surface scales smaller than Rmin do not contribute significantly to scattering. In other words, the effective scattering surface may be considered to be smoothed or “low-pass filtered” with a cutoff frequency κmax equal to the reciprocal of the minimum scale, Rmin, chosen as the smallest scale that is effective in the scattering process. Hagfors [1970] invoked this filtering idea to interpret the width parameter C, thereby circumventing the objection that a surface with exponential correlation function has infinite RMS slope. He argued that the filtering has no significant effect on the integral (4) leading to σ°H(θ). That is, filtering the surface has a negligible effect on the functional form of σ°H(θ), though it produces a meaningful surface model by rendering finite the surface RMS slope.

[23] The physical principle behind the low-pass filtering of the surface is that from a scattering point of view, surface roughness on scales that are small with respect to the EM wavelength do not contribute significantly to scattering as compared to roughness on scales comparable in size to or greater than λ. Assume, for example, an otherwise smooth surface with undulations on both vertical and horizontal scales much smaller than the wavelength. It is well known that such a surface scatters as though it were a smooth plane when illuminated by radiation of sufficiently long wavelength. On physical grounds, what the wave responds to is not the mathematical surface with its fine structure much smaller than a wavelength but, rather, some smoothed version of the surface that is a function of the original surface statistics and the EM parameters.

[24] The surface RMS slope of a filtered surface with exponential correlation function is given in Appendix A:

equation image

Choosing κmax = 1/Rmin as a, where a = (4π)/(λequation image) = λC−1/hrms2, yields the customary interpretation that Cequation image is equal to the RMS slope of the surface. This particular choice for Rmin corresponds to a fraction x = 0.26, where x is the ratio of the area under the curve of Figure 3 within the interval [0, Rmin] to the total area. For the purpose of filtering the surface and interpreting C, it is also plausible to argue, for example, that x = 0.1 or x = 0.05 would be more appropriate choices, in which cases Rmin becomes 0.53/a or 0.36/a, respectively, and consequently the RMS slope becomes 1.37Cequation image or 1.67Cequation image.

[25] In order to illustrate the significance of the above analysis, we use lunar backscatter data acquired at radar wavelengths λ = 0.68 m and λ = 0.23 m. Hagfors [1970] reported that σ°H(θ) provides an excellent fit for data obtained at these two particular wavelengths with C equal to 95 and 65 at 0.68 and 0.23 m, respectively. Note that the change in the value of C results from a change in λ as well as a change of the subradar point due to Earth-Moon libration [Simpson, 1976]. In other words, this particular example comprises two distinct surfaces at the subradar point observed at two different wavelengths. Three interpretations of the RMS slope are shown in Table 1, each with a different choice of the spatial cutoff frequency κmax = 1/Rmin corresponding to a specified area fraction x under the curve within the interval [0, Rmin]. The values given in Table 1 illustrate the uncertainty inherent in the inferred values of the surface RMS slope when σ°H(θ) is used to fit observational data. This lack of specificity arises from the indeterminate relationship between C and the RMS slope given the weak constraints on the choice of Rmin.

Table 1. Inferred RMS Slope From Two Lunar Scattering Experiments at Different EM Wavelengths and Distinct Lunar Subradar Pointsa
Surface ModelExpressionInferred RMS Slope
  λ = 0.68 m, C = 95λ = 0.23 m, C = 65
  • a

    The width parameter C is obtained by fitting the observed RCS versus angle measurements. For Hagfors' model, x denotes the fraction area under the curve in Figure 3 within the interval [0, Rmin]. For the fractal-based model, R is a specified reference horizontal scale at which the RMS slope is estimated. This table shows the dependence of the inferred RMS slope on the choice of x for the Hagfors model. In the fractal case the different values reflect the intrinsic scale dependency of the model rather than an outcome of a reasonable, albeit somewhat arbitrary, choice. When R is set to λ, it can be inferred that the C = 65 surface is rougher. Setting R to 1 m shows that the C = 95 is rougher if we compare the two surfaces at a specific horizontal scale.

Exponential correlation function   
   x = 0.26Cequation image5.9°7.1°
   x = 0.11.37Cequation image9.6°
   x = 0.051.67Cequation image9.7°11.7°
Fractal surface, H = 1/2   
   R = λequation imageCequation image7.3°
   R = 1 mequation imageCequation image3.9°

4. Hagfors' Law and Energy Conservation

[26] Assume a surface which is illuminated uniformly by normally incident plane wave EM radiation with an incident power per unit area of Pi. The resulting power dPr scattered by the surface into the solid angle dΩs is given by

equation image

where σ°bi is the specific bistatic RCS, A is the area of the surface, r is the radius of the sphere on which the received power density is calculated, and r2dΩs is the area subtended at distance r by the solid angle dΩs = sin(θs)dθsdϕs. Rewriting Pi as P/A, where P is the total incident power, and integrating (10) we obtain

equation image

where Pr is the total power received on a distant hemisphere covering the surface, and equation imagei and equation images are the incident and scattered wave polarization vectors, respectively. The summation in (11) includes two terms: one which represents the expected sense polarization component of the received power, while the other represents the orthogonal component. For a perfectly conducting surface, there is neither loss nor transmission of power through the surface. Given this and ignoring any surface waves that may be induced by the incident EM radiation, the total power received above the surface must equal the total power incident. Hence the condition of energy conservation for perfectly conducting surfaces is that

equation image

[27] Assume that the Kirchhoff approximation is used with a Gaussian surface model with an exponential correlation function. Evaluation of expression (3) for σ°bi with θi = 0° yields,

equation image

which is the bistatic version of σ°H(θ), and the polarization information is captured by factor F which is equal to 4 for the sum of the two received polarizations. Figure 4 shows the function f(C), which is equal to the ratio Pr/P when (11) is evaluated using expression (13) for the bistatic RCS with ρ = 1. As is evident from Figure 4, the bistatic version of σ°H(θ) does not conserve energy. In fact, as λ goes to zero and hence C goes to zero, the total scattered power in all directions above the surface becomes zero. Barrick [1970, p. 648], who used a one-dimensional calculation, commented on this disappearance of total scattered power in the high-frequency limit saying, “The resulting scatter solution defies conservation of power in the high-frequency limit.”

Figure 4.

Evaluation of f(C): the ratio of the total power received on a distant covering hemisphere to the total incident power as a function of parameter C. Curve is calculated using (11), where the bistatic RCS is given by (13). Plot illustrates that the bistatic equivalent of Hagfors' law does not conserve energy.

[28] The reason for this “disappearance” is that the spectrum of a surface with exponential correlation function decreases relatively slowly with spatial frequency, resulting in significant high-frequency components and small-scale roughness. As the EM wavelength λ decreases, and because of the increased roughness at scales comparable in size to λ, the surface scatters more diffusely with the result that KA, which generally accounts only for quasi-specular scattering, predicts less scattered power. In other words, as λ tends to zero, the average effective radius of curvature of the surface progressively diminishes with the result that the application of KA on which σ°H(θ) is based goes well beyond the limits of its validity, as discussed briefly in section 2. This argument is not restricted to surfaces with an exponential correlation function; it applies to any surface model which has roughness on arbitrarily small scales [Sultan-Salem and Tyler, 2004].

[29] In scattering from rough surfaces, it may be useful to consider the diffuse scattered field as the scattering not accounted for by use of KA. This includes, for instance, multiple scattering, volume scattering, and scattering from roughness on scales small compared to the EM wavelength λ. Although it is true that scattering from small-scale roughness, always as compared to λ, is not as considerable as from large-scale roughness, the diffuse scattering from a surface with substantial small-scale roughness can be significant. It is the combination of the overall distribution of roughness on a particular surface and the probing EM wavelength that determines the scattering behavior of such a surface. Appendix B discusses how a diffuse component, when added to a KA-based scattering law to account for unmodeled scattering processes, can be constrained using a conservation of energy argument.

5. Comparison With a Fractal-Based Scattering Law

[30] As mentioned in section 1, fractal geometry has come to play an important role in surface modeling. One widely used fractal model for study of EM wave scattering is the “fractional Brownian motion model” [e.g., Franceschetti et al., 1999; Shepard and Campbell, 1999]. The FB surface is defined in terms of stationary Gaussian surface increments whose variance over space is the same as that of Brownian motion over time,

equation image

where H ∈ [0, 1] is the Hurst exponent, R is the horizontal separation between two points on the surface as before, and s is the RMS slope at a horizontal separation, or scale, of unity. The RMS slope, S, at a reference horizontal scale, R, is given by

equation image

[31] The value of the Hurst exponent controls the variation of surface roughness with horizontal scales. Surface profiles with values of H approaching zero are rough on small scales but are relatively smooth on large scales. Conversely, surface profiles with H approaching unity are equally rough on all horizontal scales [Shepard and Campbell, 1999]. The condition H = 1 results in a self-similar surface for which vertical relief increases at the same rate as the horizontal scale at which it is measured. The four profiles in Figure 5 illustrate the dependence of surface roughness on H, namely, that increasing H increases the height variations on large scales.

Figure 5.

Four fractional Brownian profiles with H = 0.2, 0.4, 0.6, and 0.8. Profiles have zero mean but are offset vertically for clarity. All profiles have unity slope at a horizontal scale of one unit, i.e., S = 1 at R = 1. The effect of increasing H is to increase height variations on large scales.

[32] Shepard and Campbell [1999] and Campbell and Shepard [2003] studied the coherent component, defined by (1), of near-normal incidence backscatter signals, starting from a standard scalar formulation of Huygens principle for radiation from an aperture antenna. Employing the FB fractal model, KA, and the averaging method of expression (2), Franceschetti et al. [1999] developed a fractal-based scattering law (incorrectly characterized as incoherent by Campbell and Shepard [2003]) for which the specific backscatter RCS is given by

equation image

Figure 6 illustrates the behavior of fractal-based backscatter RCS for several combinations of H and λ-scale S. The specific backscatter RCS at normal incidence is

equation image

where Γ denotes the gamma function.

Figure 6.

Fractal-based scattering laws with ρ = 1. (a) RMS slope at scale λ is equal to 0.1. (b) RMS slope at scale λ is equal to 0.3. Note that increasing the RMS slope results in widening the scattering lobe as the surface roughens and that the behavior at θ = 0, given by (17), is reversed for varying H.

[33] For H = 1/2, the resultant RCS from (16) has the same functional form as σ°H(θ) given by (5), but with the difference that C = [λ/(2πR)]2S−4. It follows that the RMS slope S at horizontal scale R for H = 1/2 is given by

equation image

Note that the RMS slope of this particular fractal surface is proportional to Cequation image rather than Cequation image. Moreover, the inferred RMS slope here depends on the relationship of the EM wavelength to the reference scale, whereas the specific horizontal scale linked with the RMS slope inferred from Hagfors' model, and classical models in general, is not explicitly available. On the basis of the H = 1/2 fractal-based scattering law, we can calculate, for the two lunar surfaces discussed in section 3, additional RMS slope estimates using (18).

[34] For illustration we use two reference scales, R = 1 m and R = λ, as given in Table 1. Now we ask the question: Which of the two surfaces is rougher granted that the functional form of σ°H(θ) is an adequate model for the observed scattering behavior? As is clear from rows 1–3 of Table 1, it can be inferred on the basis of σ°H(θ), despite the indefiniteness, that the C = 95 surface is smoother than the C = 65 surface at the respective EM wavelengths used to probe each surface. Setting R to λ produces a similar inference since the λ-scale RMS slope of the C = 65 is larger than that of the C = 95 surface. In addition, the fractal-based interpretation, because of its explicit scale dependency, can provide an answer to the question as to which surface is rougher at any specific horizontal scale. Setting R to 1 m shows that the C = 65 surface is smoother if the comparison of roughness is done at the reference horizontal scale of 1 m. Since from (15) the RMS slope S at scale R is equal to s/equation image when H = 1/2, and since s, the RMS slope at unity horizontal separation, is smaller for the C = 65 surface as is evident from row 5 of Table 1, then it follows that the C = 65 surface is smoother at any specified horizontal scale lying within the range over which the given fractal characterizations are valid.

[35] The frequency behavior of the fractal-based laws parameterized by H and s is similar to the behavior of σ°H(θ) discussed in section 3. At normal incidence, σ°, which for a given s is proportional to λequation image as given in (17), decreases as the wavelength diminishes as a result of the increasing effectiveness of λ-scale surface roughness in increasing the fraction of scattered power that is removed from the quasi-specular direction. It is straightforward to show that the fractal-based scattering laws also fail to conserve energy, and that the left-hand side of (12) increasingly becomes less than unity as the RMS slope of the FB surface increases.

[36] Setting the upper limit of (16) to ∞ results in an overestimation in the value of the integral for the surface RCS when the physical size of illuminated surface is finite and equal to Rmax. As discussed in connection with σ°H(θ), this does not imply that fractal-based laws generally overestimate actual RCS values because, being derived on the basis of tangent plane approximation, they also do not account for all the power that is actually scattered. In order to limit the overestimation error, which is an issue when the radar resolution cells are comparable in size to the EM wavelength, the illuminated surface dimensions must exceed some value that is a function of λ and the surface parameters.

[37] Campbell and Shepard [2003] estimate the surface dimension Rmax by requiring that the maximum backscatter RCS value, σ°(0) given by (17), not exceed 4π(πRmax2)ρ/λ2, which is the normal specific backscatter RCS of a smooth disk of radius Rmax. Therefore they set the lower limit of Rmax to (λ/2π)equation image. We provide an estimate following the same approach as that leading to (7) for a Gaussian surface with exponential correlation function. For an FB surface defined in terms of the parameters s and H, and requiring the overestimation error at normal incidence to be less than 10%, we find

equation image

where the decimal constants are determined numerically.

[38] In order to illustrate the relevance of inequalities (7) and (19) to scattering experiments, we apply (19) to the analysis of surface and subsurface scattering in MARSIS, following Campbell and Shepard [2003]. According to Orosei et al. [2003] and on the basis of Mars Orbiter Laser Altimeter (MOLA) data, the histogram of the Hurst exponent of the Martian surface can be fit by a Weibull distribution with a mean H = 0.7 and a standard deviation of 0.19. This result is restricted to the “monofractal” profiles for which H appears constant with the horizontal scales used in the calculations, starting with the 300-m separation of MOLA sampling. The 300-m RMS slope of those topographic profiles that exhibit monofractal behavior has a mean of 0.038 and a standard deviation of 0.039.

[39] Using s = SR1−H, S = 0.038 at R = 300 m corresponds to s = 0.21 when H = 0.7. The longest carrier wavelength used in MARSIS is 167 m while the shortest is 60 m [Picardi et al., 1999]. Given H = 0.7, s = 0.21, and a maximum overestimation error of 10 percent, expression (19) leads to minimum sizes for the radar resolution cells of 321 and 1384 m when λ = 60 and 167 m respectively. Table 2 gives the minimum value of the surface extent Rmax for a 10% error and four different combinations of fractal surface parameters H and s. As is evident from Table 2, the smoother the surface and the larger the wavelength, the larger the resolution cells must be in order to maintain the overestimation error, resulting from extension of the upper limit of integral (16) to ∞, below 10%. To put the numbers in Table 2 into perspective, note that in MARSIS the resolution in the along-track direction is 5000–9000 m [Picardi et al., 1999]. The actual size of MARSIS resolution cells depends on the EM wavelength, orbit parameters and the location of the resolution cell with respect to the subradar point.

Table 2. Minimum Value of the Surface Size Rmax Such That the Overestimation Error in the Surface RCS at Normal Incidence Is Less Than 10%a
 Slope S at R = 300 mMinimum Value of Rmax, m
λ = 60 mλ = 167 m
  • a

    The wavelengths 60 and 167 m are the minimum and maximum employed in MARSIS. As is evident from this table, the smoother the surface and the larger the wavelength, the larger the resolution cells must be to maintain the overestimation error, which results from extending the upper limit of integral (16) to ∞, below the specified value of 10%.

H = 0.7, s = 0.212.17°3211384
H = 0.7, s = 0.11.04°9253993
H = 0.6, s = 0.211.23°9245091
H = 0.6, s = 0.10.59°318317,531

6. Comments and Conclusions

[40] The form σ°H(θ) was originally derived for a classical surface model with KA boundary conditions and exponential correlation function, linearized for purpose of evaluation. Because σ°H(θ) provides a good match with observational data, it is widely used for the analysis of backscatter data from planetary surfaces. The applicability of σ°H(θ) is constrained by the limitations introduced by use of a classical surface model, reliance on KA, and the method of integration leading to σ°H(θ). In contradiction to Barrick [1970], who emphasized the mathematical and physical inadequacy of surface models with a Gaussian height distribution and an exponential correlation function, we trace the limitations to the approximations involved in the derivation of σ°H(θ).

[41] The surface RMS slope parameter inferred on the basis of σ°H(θ) is not explicit in its dependence on scale, as is also the case for other classical scattering laws. This is an undesirable feature due to the known scale-dependent height and slope properties of natural surfaces. Moreover, the relationship between the width parameter C and the surface RMS slope cannot be determined precisely. Although the filtering argument used in the interpretation of C is physically sound, there is a range of plausible choices for the cutoff scale, leading to an indefinite relationship between C and the surface RMS slope.

[42] Hagfors' law does not conserve energy. The Kirchhoff approximation, which is the basis of σ°H(θ), involves approximation of the local surface with an infinite tangent plane in order to estimate the surface fields due to the incident wave. As a result of this physical approximation, as well as other mathematical approximations needed to facilitate the evaluation of the scattering integral, KA fails to adequately represent the scattering behavior of the surface in its entirety. The outcome is a violation of the principle of conservation of energy if only a KA-based scattering component is included in the analysis of observations dominated by nonspecular scattering. It must be emphasized that all KA-based scattering laws appear to have this feature, though the problem is most profound with surface models with substantial small-scale roughness. The spectrum of a Gaussian surface with an exponential correlation function, which is the surface model of σ°H(θ), decreases slowly with spatial frequency indicating the presence of significant fine structure scales and a consequent difficulty for KA to model the whole scattering process.

[43] Applying KA to an FB surface model with H = 1/2 results in the same functional form as σ°H(θ). Reliance on KA to obtain fractal-based scattering laws leads to manifest similarities between the limitations of σ°H(θ) and the fractal-based laws. Both σ°H(θ) and fractal-based laws fail to conserve energy; neither accounts for the totality of scattered energy. Also, the accuracy of RCS values given by both models requires that the illuminated surface size exceeds some limit that is a function of λ and the surface parameters.

[44] The apparent advantages of the fractal-based laws over σ°H(θ) and other classical scattering laws are that (1) they have an additional roughness parameter H that specifies the physical variation in surface roughness with horizontal scale and (2) they provide estimates for surface RMS slope for which the dependence on scale is explicit. The additional degree of freedom in parameter H expands the range of the observed variations of surface RCS with angle that can be addressed. The scale information makes the inferred surface parameters more useful descriptors of the observed characteristics of natural surfaces. Further investigation is needed to better understand the fractal-based laws, their strengths, and their limitations.

Appendix A:: Derivation of the RMS Slope

[45] We present here the derivation of the RMS slope of a Gaussian surface with exponential correlation function following Hagfors [1970]. As mentioned in the text, the exponential correlation function is given by Cf(R) = exp(−R), with an added subscript ‘f’ to distinguish the correlation function from width parameter C. The power spectral density of the surface S(κ) is given by

equation image

Assume that the surface is subjected to an ideal low-pass filter with maximum frequency κmax. The correlation function of the filtered surface is given by

equation image

The term on the right is a normalization factor to ensure that C*f(0) = 1. The adirectional slope variance (SV) is equal to −2h*rms2[∂2C*f(R)/∂R2]∣R=0 [see Ogilvy, 1991, p. 21], where h*rms is the RMS height of the filtered surface and is given by

equation image

Therefore the variance of slope is

equation image

[46] Assuming κmaxλ ≫ 1, SV = hrms2κmax, and the RMS slope is the square root of SV.

Appendix B:: Constraining Diffuse Scattering

[47] The diffuse σ°(θ) ∝ cosn(θ) term, which is routinely added to σ°H(θ) to account for the nonspecular scattering, can be quantified using the conservation of energy principle given a set of assumptions. For this, assume that the bistatic form of the diffuse component with expected sense of polarization is given by Kecosequation imagei)cosequation images), and the diffuse component of the orthogonal polarization sense is given by Kocosequation imagei)cosequation images). These are the bistatic forms of the cosine function in common use for analyzing diffuse backscatter data. Inserting these bistatic RCS formulas into (11) gives Ke/(ne + 2) + Ko/(no + 2). When Hagfors' bistatic form given by (13) is used with the cosine terms, the conservation of energy criterion given by (12) dictates that

equation image

where f(C) in Figure 4 can be approximated by equation image/(equation image + 1.675).

[48] Now we apply the above to the backscatter case. For the expected polarization component of the received power, it is common to use the following functional form to represent the variation of the specific backscatter cross section with angle:

equation image

[e.g., Campbell, 2002; Harmon et al., 1992]. The first term is σ°H(θ) and the cosne(θ) term is the monostatic version of the bistatic diffuse component discussed above. The orthogonally polarized component is modeled by image alone, as KA does not predict depolarization in the monostatic case.

[49] Despite being derived on the basis of a specific bistatic observational setting, the condition given by (B1) is a general constraint for scattering from perfectly conducting surfaces that is well-modeled by the linear combination of σ°H(θ) and a cosine diffuse component. The same condition applies for dielectric surfaces under the assumption that the partially reflecting nature of the surface is reasonably approximated by the multiplicative factor ρ. Given this, the interpolation problem of fitting backscatter RCS versus angle data is one of finding the parameters C, Ke, ne, Ko, no under the constraint given by (B1), in addition to the scaling parameter ρ. In some cases, the σ°H(θ) component in (B2) might be multiplied by an additional factor to take into account the ratio of the surface area scattering quasi-specularly to the diffusely scattering area; the same factor should then scale f(C) in (B1).

[50] If a KA-based scattering law other than σ°H(θ) is used to model quasi-specular scattering, then the integral of the law, given by (11), is used instead of f(C). Using (3) and (4), it can be shown that the bistatic equivalent of a backscatter law, derived using the KA formalism presented in this paper, can be obtained by substituting in the monostatic formula θ by θ/2 and λ by λsec(θ/2). This last point is a mathematical observation on the basis of (3) and (4) that is similar to, though different in essence, from the result of Kell [1965], where he derived the bistatic RCS from monostatic measurements.

[51] It is important to note that the above analysis is theoretical and assumes, in addition to the representation of the lost energy by the scaling parameter ρ, that physically meaningful scattering models are employed that yield an acceptable goodness of fit or statistical significance measures. Studies involving application to observational data will help in understanding how the conservation of energy constraint may be employed in actual modeling of scattering from planetary surfaces.

Acknowledgments

[52] The authors gladly acknowledge helpful comments and suggestions by two anonymous reviewers. This work was supported by the NASA Planetary Geology and Geophysics Program under NASA grant NNG04GG00G.

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